A Mean Field Game Approach to Scheduling in Cellular Systems

Abstract

There has been much work done on designing cellular scheduling algorithms. These algorithms are set up in the manner of a direct mechanism in which the queues reveal their states (backlog), and the scheduler chooses an allocation. In policies such as longest queue first (LQF), scheduling can be shown to yield short queue lengths. However, these algorithms are reliant on truthful declarations of state. Our goal in this article is to determine if a Vickrey–Clarke–Groves (VCG)-type mechanism (a second price auction) that elicits truthful value will also possess the desired LQF-like behavior in a system of dynamically evolving queues. Our approach is to use the concept of a mean field game under which at each time instant, a queue chooses its bid as a best response to its belief that the bids of the others will be drawn independently from a common bid distribution. If this best response is itself a sample from the belief bid distribution, a mean field equilibrium (MFE) is said to exist. We show the existence of the MFE and find it using its structure that the results of the allocation policy are the same as LQF. Thus, the desired LQF-like behavior arises naturally under the second-price auction mechanism. We also present results on the accuracy of the model as the number of agents (queues) becomes large. Finally, using simulations with a large number of queues, we show that computation of the MFE is straightforward.